algorithms Article Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization Wei Li 1,2 1 School of Computer Science and Engineering, Xi’an University of Technology, Xi’an 710048, China; [email protected] 2 Shaanxi Key Laboratory for Network Computing and Security Technology, Xi’an 710048, China Received: 15 January 2019; Accepted: 27 February 2019; Published: 5 March 2019 Abstract: The standard covariance matrix adaptation evolution strategy (CMA-ES) is highly effective at locating a single global optimum. However, it shows unsatisfactory performance for solving multimodal optimization problems (MMOPs). In this paper, an improved algorithm based on the MA-ES, which is called the matrix adaptation evolution strategy with multi-objective optimization algorithm, is proposed to solve multimodal optimization problems (MA-ESN-MO). Taking advantage of the multi-objective optimization in maintaining population diversity, MA-ESN-MO transforms an MMOP into a bi-objective optimization problem. The archive is employed to save better solutions for improving the convergence of the algorithm. Moreover, the peaks found by the algorithm can be maintained until the end of the run. Multiple subpopulations are used to explore and exploit in parallel to find multiple optimal solutions for the given problem. Experimental results on CEC2013 test problems show that the covariance matrix adaptation with Niching and the multi-objective optimization algorithm (CMA-NMO), CMA Niching with the Mahalanobis Metric and the multi-objective optimization algorithm (CMA-NMM-MO), and matrix adaptation evolution strategy Niching with the multi-objective optimization algorithm (MA-ESN-MO) have overall better performance compared with the covariance matrix adaptation evolution strategy (CMA-ES), matrix adaptation evolution strategy (MA-ES), CMA Niching (CMA-N), CMA-ES Niching with Mahalanobis Metric (CMA-NMM), and MA-ES-Niching (MA-ESN). Keywords: multimodal optimization problems; multi-objective optimization; matrix adaptation evolution strategy; non-dominated sorting 1. Introduction Many problems from the real world are classified as optimization problems. Some optimization problems that have one global solution are called single modal optimization problems, while others that have multiple global and local optima are known as multimodal optimization problems. Traditional evolutionary algorithms (EAs) are effective at converging to a single global optimum because of the global selection strategy used. However, it is inappropriate for EAs to solve multimodal optimization problems. In order to overcome the weakness, niching techniques are incorporated into EAs, such as differential evolution [1,2], particle swarm optimization [3], the covariance matrix adaptation evolution strategy (CMA-ES) [4], self-adaptive niching CMA-ES [4], and genetic algorithm [5], to solve multimodal optimization problems. The representative niching strategies include crowing [6], restricted tournament selection [7], fitness sharing [8], clearing [9], and speciation [5]. The covariance matrix adaptation evolution strategy (CMA-ES), which was proposed by Hansen and Ostermeier [10], is one of the popular optimization algorithms for solving unconstrained Algorithms 2019, 12, 56; doi:10.3390/a12030056 www.mdpi.com/journal/algorithms Algorithms 2019, 12, 56 2 of 23 real-parameter optimization. Different from other optimization algorithms, CMA-ES makes use of two evolution paths to realize exploitation and exploration during the search process. The two evolution paths are the learning of the mutation strength and the rank-1 update of the covariance matrix, respectively. The self-adaptively updated covariance matrix, which uses evolution path information, can be considered a time series prediction of the evolution of the parent [11]. Since CMA-ES employs the covariance matrix to exploit the information from the previous and current generations, it has attracted broad investigation in recent years. However, compared with other classical algorithms, such as DE or PSO, CMA-ES is slightly more complicated, because it has two evolution paths and an update of the covariance matrix. To simplify the standard CMA-ES, Beyer and Sendhoff proposed the matrix adaptation evolution strategy (MA-ES) [11], in which one of the evolution paths (namely the p-evolution path) is dropped, and the covariance matrix (namely the C matrix) is discarded. The experimental results in [11] show that the MA-ES exhibits a similar performance as the CMA-ES, which considers both standard population sizes (l < N) and large population sizes (l = O (N2)). The MA-ES only simplifies the CMA-ES. The performance of the MA-ES is basically the same as that of the CMA-ES. Therefore, the MA-ES is a robust local search strategy that efficiently solves unimodal optimization problems. It is unable to find multiple solutions in multimodal problems because of the designed parameters and updating rules [12]. At present, to the best of our knowledge, no work has been reported on utilizing the MA-ES to solve the multimodal problems. There has been an effort to provide two versions of CMA-ES, which are called the niching covariance matrix adaptation evolution strategy and the self-adaptive niching CMA-ES, respectively, for solving multimodal problems [13,14]. The two improved versions of the CMA-ES introduced niching strategies that can maintain the population diversity and realize parallel convergence within some subpopulations to obtain multiple good solutions. However, the performance of the two improved versions of CMA-ES is highly sensitive to niching parameters. So far, some works have been done to convert a multimodal optimization problem (MMOP) to a multi-objective optimization problem (MOP) [15–19]. The advantage of transforming an MMOP into an MOP is that it is unnecessary to use the problem-dependent niching parameters. However, the prerequisite for MOP is objective confliction, which makes it difficult to transform an MMOP into a multi-objective optimization problem [17]. To address this issue, this paper proposes an improved algorithm based on the MA-ES called the matrix adaptation evolution strategy with multi-objective optimization algorithm (MA-ESN-MO). The main contributions of this paper are summarized as follows: 1. The MMOP is transferred into two objective optimization problems with strong objective confliction; the advantage of multi-objective optimization can be fully used to ensure the diversity of the population. 2. The information of the population landscape and the fitness of the objective function are employed to construct two conflicting objective functions instead of utilizing classical niching strategies. Moreover, the archive is employed to save better individuals, which are helpful to ensure the convergence of the algorithm. In this manner, the exploration and exploitation abilities of the algorithm are balanced effectively. 3. The population is divided into several subpopulations in MA-ES instead of using one population in CMA-ES. In this way, the algorithm can explore and exploit in parallel within these subpopulations to find multiple optimal solutions for the given problem. Moreover, the niching method is employed to improve the diversity of the population. 4. Systematic experiments conducted to compare the algorithms including CMA-ES, MA-ES, CMA-ES-Niching (CMA-N), CMA-ES-Niching-MO (CMA-NMO), CMA-ES Niching with Mahalanobis Metric [20] (CMA-NMM), CMA-NMM-MO, MA-ES-Niching (MA-ESN), and MA-ESN-MO on CEC2013 multimodal benchmark problems [21] are described. CMA-NMO and CMA-NMM-MO are obtained by introducing the proposed method into CMA-ES and CMA-NMM. The experimental results show that the proposed method is promising for solving multimodal optimization problems. Algorithms 2019, 12, 56 3 of 23 The rest of this paper is organized as follows. In Section2, the related work on multimodal optimization problems is reviewed. Section3 introduces variants of CMA-ES and the framework of MA-ES. The proposed matrix adaptation evolution strategy with the multi-objective optimization algorithm (MA-ESN-MO) is presented in Section4. Section5 reports and discusses the experimental results. Finally, the conclusions and possible future research are drawn up in Section6. 2. Related Work Many evolutionary algorithms (EAs) can effectively solve single-objective optimization problems that involve only one optimal solution. However, they are unable to perform well on multimodal optimization problems because of their poor population diversity preservation. To address the issue, sufficient works have been done over the past decades. The strategies on improving the EAs fall into three categories [22]. Niching is an effective method that is used to find and preserve multiple stable niches for multimodal optimization problems. Classical niching techniques include crowding, fitness sharing, speciation, clearing, and restricted tournament selection. The two classic crowding strategies are deterministic crowding and probabilistic crowding. The deterministic crowding can effectively solve the problem of the replacement error, which is the main disadvantage of crowding, while probabilistic crowding utilizes the probabilistic selection to prevent the loss of niches with lower fitness or the loss of local optima [23–25]. Both the fitness sharing strategy and speciation divide the population into several subpopulations according to the similarity of the individuals, which can form and maintain the stable niches [5,8].